以下是原文
For three-dimensional navigation signals from at least four synchronized transmitters need to be received. The three independent time differences generate three different rotational hyperboloids. Rotational hyperboloids are curved surfaces. Two of them intersect in a curved line which in turn intersects with the third hyperboloid in a point corresponding to the unknown three-dimensional user position.
If there are more transmitters available, the user can select the best set of three (four) that provide two hyperbolas (three rotational hyperboloids) intersecting as close as possible under a right angle(s). The remaining transmitters can then be used to check for errors and/or ambiguous solutions, since with curved lines and surfaces there can be more than one intersection point.
Hyperbolic navigation systems were first implemented as ground-based navigation systems operating in the medium and long-wave radio frequency spectrum like LORAN, DECCA or OMEGA. Since the transmitters are located on the Earth's surface, the geometry of the problem does not allow a three-dimensional navigation. These systems only measure the longitude and latitude reliably. To measure the altitude, one of the transmitters should be located above or below the user's receiver or at least out of the user's horizon plane.
Ground-based radio-navigation systems use relatively low frequencies of the radio spectrum to achieve a large radio range and avoid undefined skywave (ionospheric) propagation at the same time. For example, OMEGA uses the frequency range between 10 and 14kHz to achieve world-wide coverage with just 8 (eight) transmitters!
Long-wave radio-navigation systems were designed when digital computers were not readily available yet: two-dimensional navigation with fixed transmitter sites only requires a minimum of computations to be performed by the user. The families of hyperbolas for each transmitter pair can be directly plotted on maps, including corrections for known propagation anomalies.
One of the first applications of artificial satellites was radio navigation. Obviously artificial satellites need radio navigation themselves, to evaluate the performance of the rocket carrier and determine the final satellite's orbit. On the other hand, the space environment is an ideal place for navigation transmitters, since a large radio range can be achieved at VHF and higher frequencies where the propagation of radio waves is predictable and the influence of the always-changing ionosphere is marginal. Finally, the location of navigation transmitters in space can be chosen so that three-dimensional navigation is possible everywhere on the Earth's surface.
Since initially the satellites could only be launched in low-earth orbits, the first navigation satellites were launched in low, 1000km altitude, polar orbits, like the American TRANSIT satellites or the soviet equivalent TSIKADA. Since a satellite in a low-Earth orbit is quickly moving along its orbital track, a single satellite may be used for position determination. While even a simple crystal-controlled user's clock is sufficiently accurate for a few minutes, the satellite significantly changes its position on the sky and this is roughly equivalent to several navigation transmitters at several different sites along the orbital track.
In practice the user simply measures the Doppler shift on the satellite's signal for a certain period of time and computes his unknown position from the result of this measurement and the satellite's orbital data. Although a single satellite is required for position determination, these systems usually have from six (TRANSIT) up to twelve satellites to improve the coverage, since a low-Earth orbit satellite is only visible for a limited amount of time for a user located on the Earth's surface. Since the ionosphere still has some effect on VHF and UHF radio waves, both American and Soviet satellites transmit on two frequencies around 150MHz and around 400MHz. The actual channel frequencies are kept in the precise ratio 3/8 and the transmitters are kept coherent to allow for ionospheric corrections.
The drawbacks of low-Earth orbit navigation satellites are that the user may have to wait for a satellite pass and even then the measurement takes several minutes. Finally, the user velocity, both magnitude and direction, must be known and compensated-for in the position computation. To allow an almost instantaneous position determination more satellites are required. If a user has at least four visible satellites in different parts of the sky, he can compute his three-dimensional position instantaneously, without having to wait for the satellites to move across the sky.
In order to limit the number of satellites required, these have to be launched to higher orbits. Such satellite navigation systems are the American GPS and the soviet GLONASS that should achieve world-wide coverage with 24 satellites each when completed. Both systems should provide at least four visible satellites in any part of the world including in-orbit spares and a suitable distribution of the visible satellites on the sky to allow a three-dimensional navigation.
Finally, one should notice that satellite navigation systems require a large amount of computations to be performed by the user. The satellites continuously change their positions, so no hyperbolas could be plotted on maps. Three-dimensional navigation is even more demanding, so that a digital computer is absolutely necessary. Maybe this explains why satellite positioning only became popular a few years ago: although navigation satellites were available for more than 30 years, inexpensive computers were not! |
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